Answer: Certainly, the deeper the better in terms of the root crops as you have surmised. Would it be possible to loosen the soil below the bed and thus set the wire a bit deeper in one or more areas specifically intended to accommodate them? Alternatively, you could mound the soil higher than the actual edges of the beds. If the sides are sloped steeply enough there will be minimal erosion. Lastly, you could try growing some of the smaller varieties of carrots, for instance, to fit better in shallower soil and consider using a tall cylinder of wire mesh for the potatoes. I hope this gives you some ideas.
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